Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Conference Object Measure on Time Scales With Mathematica(Springer Verlag, 2006) Ufuktepe, Ünal; Yantır, AhmetIn this paper we study the Lebesgue Delta-measure on time scales. We refer to [3, 4] for the main notions and facts from the general measure and Lebesgue Delta integral theory. The objective of this paper is to show how the main concepts of Mathematica can be applied to fundamentals of Lebesgue Delta- and Lebesgue Delta- measure on an arbitrary time scale and also on a discrete time scale whose rule is given by the reader. As the time scale theory is investigated in two parts, by means of alpha and rho operators, we named the measures on time scales by the set function DMeasure and NMeasure respectively for arbitrary time scales.Article Citation - WoS: 3Citation - Scopus: 3Co-Coatomically Supplemented Modules(Springer Verlag, 2017) Alizade, Rafail; Güngör, SerpilIt is shown that if a submodule N of M is co-coatomically supplemented and M/N has no maximal submodule, then M is a co-coatomically supplemented module. If a module M is co-coatomically supplemented, then every finitely M-generated module is a co-coatomically supplemented module. Every left R-module is co-coatomically supplemented if and only if the ring R is left perfect. Over a discrete valuation ring, a module M is co-coatomically supplemented if and only if the basic submodule of M is coatomic. Over a nonlocal Dedekind domain, if the torsion part T(M) of a reduced module M has a weak supplement in M, then M is co-coatomically supplemented if and only if M/T (M) is divisible and TP (M) is bounded for each maximal ideal P. Over a nonlocal Dedekind domain, if a reduced module M is co-coatomically amply supplemented, then M/T (M) is divisible and TP (M) is bounded for each maximal ideal P. Conversely, if M/T (M) is divisible and TP (M) is bounded for each maximal ideal P, then M is a co-coatomically supplemented module.Article An Application With Webmathematica(Springer Verlag, 2003) Ufuktepe, ÜnalThere have been many technological dawns in the last 30 years, during which the desktop computer and the Internet have been developed. The importance of Internet in education, particularly using its Web is a well-recognized fact. A wealth of resources and techniques now exist which serve as a source both for exciting examples of new teaching practices, as well as easily accessible methods for adoption into various formats of teaching and learning. Internet technology allow teachers and students keep up with their minds. It let them try their ideas as soon as they come up with them. Generally, students appreciate the convenience, choice, and flexibility that an online courses offers. Instructional designers value the standardized framework and flexibility. WebMathematica is a web-based technology developed by Wolfram Research that allows the generation of dynamic web content with Mathematica. With this technology, the distance education students should be able to explore and experiment with the mathematical concepts. In this paper we will elucidate the pedagogical issues in the application of Hamiltonian systems in the webMathematica for the distance learning environment and the shape of the future "classroom" as well as relevant educational strategies towards improving mathematics education.Article Citation - WoS: 2Citation - Scopus: 1Basic Calculus on Time Scale With Mathematica(Springer Verlag, 2003) Yantır, Ahmet; Ufuktepe, ÜnalMathematical modeling of time dependent systems are always interesting for applied mathematicians. First continuous and then discrete mathematical modeling are built during the mathematical development from ancient to the modern times. By the discovery of the time scales, the problem of irregular controlling of time dependent systems is solved in 1990's. In this paper, we explain the derivative of functions on time scales and the solutions of some basic calculus problems by using Mathematica. © Springer-Verlag Berlin Heidelberg 2003.Article Citation - WoS: 3Citation - Scopus: 3Exactly Solvable Madelung Fluid and Complex Burgers Equations: a Quantum Sturm-Liouville Connection(Springer Verlag, 2012) Atılgan Büyükaşık, Şirin; Pashaev, OktayQuantum Sturm-Liouville problems introduced in our paper (Büyükaşi{dotless}k et al. in J Math Phys 50:072102, 2009) provide a reach set of exactly solvable quantum damped parametric oscillator models. Based on these results, in the present paper we study a set of variable parametric nonlinear Madelung fluid models and corresponding complex Burgers equations, related to the classical orthogonal polynomials of Hermite, Laguerre and Jacobi types. We show that the nonlinear systems admit direct linearazation in the form of Schrödinger equation for a parametric harmonic oscillator, allowing us to solve exactly the initial value problems for these equations by the linear quantum Sturm-Liouville problem. For each type of equations, dynamics of the probability density and corresponding zeros, as well as the complex velocity field and related pole singularities are studied in details. © 2012 Springer Science+Business Media, LLC.Article Citation - WoS: 9Citation - Scopus: 10Applications of the Pseudo Residual-Free Bubbles To the Stabilization of Convection-Diffusion Problems(Springer Verlag, 2012) Şendur, Ali; Neslitürk, Ali İhsanIt is known that the enrichment of the polynomial finite element space of degree 1 by bubble functions results in a stabilized scheme of the SUPG-type for the convection-diffusion-reaction problems. In particular, the residual-free bubbles (RFB) can assure stabilized methods, but they are usually difficult to compute, unless the configuration is simple. Therefore it is important to devise numerical algorithms that provide cheap approximations to the RFB functions, contributing a good stabilizing effect to the numerical method overall. Here we propose a stabilization technique based on the RFB method and particularly designed to treat the most interesting case of small diffusion. We replace the RFB functions by their cheap, yet efficient approximations which retain the same qualitative behavior. The approximate bubbles are computed on a suitable sub-grid, the choice of whose nodes are critical and determined by minimizing the residual of a local problem with respect to L 1 norm. The resulting numerical method has similar stability features with the RFB method for the whole range of problem parameters. This fact is also confirmed by numerical experiments. We also note that the location of the sub-grid nodes suggested by the strategy herein coincides with the one in Brezzi et al. (Math. Models Methods Appl. Sci. 13:445-461, 2003). © 2011 Springer-Verlag.Article Citation - WoS: 7Citation - Scopus: 7Strongly Radical Supplemented Modules(Springer Verlag, 2012) Büyükaşık, Engin; Türkmen, ErgülZöschinger studied modules whose radicals have supplements and called these modules radical supplemented. Motivated by this, we call a module strongly radical supplemented (briefly srs) if every submodule containing the radical has a supplement. We prove that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect. Over a local Dedekind domain, srs-modules and radical supplemented modules coincide. Over a nonlocal Dedekind domain, an srs-module is the sum of its torsion submodule and the radical submodule. © 2012 Springer Science+Business Media, IncConference Object Citation - WoS: 34On the Importance of Public-Key Validation in the Mqv and Hmqv Key Agreement Protocols(Springer Verlag, 2006) Menezes, Alfred; Ustaoğlu, BerkantHMQV is a hashed variant of the MQV key agreement protocol proposed by Krawczyk at CRYPTO 2005. In this paper, we present some attacks on HMQV and MQV that are successful if public keys are not properly validated. In particular, we present an attack on the two-pass HMQV protocol that does not require knowledge of the victim's ephemeral private keys. The attacks illustrate the importance of performing some form of public-key validation in Diffie-Hellman key agreement protocols, and furthermore highlight the dangers of relying on security proofs for discrete-logarithm protocols where a concrete representation for the underlying group is not specified.Article Citation - WoS: 22Anonymity and One-Way Authentication in Key Exchange Protocols(Springer Verlag, 2013) Goldberg, Ian; Stebila, Douglas; Ustaoğlu, BerkantKey establishment is a crucial cryptographic primitive for building secure communication channels between two parties in a network. It has been studied extensively in theory and widely deployed in practice. In the research literature a typical protocol in the public-key setting aims for key secrecy and mutual authentication. However, there are many important practical scenarios where mutual authentication is undesirable, such as in anonymity networks like Tor, or is difficult to achieve due to insufficient public-key infrastructure at the user level, as is the case on the Internet today. In this work we are concerned with the scenario where two parties establish a private shared session key, but only one party authenticates to the other; in fact, the unauthenticated party may wish to have strong anonymity guarantees. We present a desirable set of security, authentication, and anonymity goals for this setting and develop a model which captures these properties. Our approach allows for clients to choose among different levels of authentication. We also describe an attack on a previous protocol of Øverlier and Syverson, and present a new, efficient key exchange protocol that provides one-way authentication and anonymity. © 2012 Springer Science+Business Media, LLC.Article Citation - Scopus: 9Sufficient Condition for Ephemeral Key-Leakage Resilient Tripartite Key Exchange(Springer Verlag, 2012) Fujioka, Atsushi; Manulis, Mark; Suzuki, Koutarou; Ustaoğlu, BerkantTripartite (Diffie-Hellman) Key Exchange (3KE), introduced by Joux (ANTS-IV 2000), represents today the only known class of group key exchange protocols, in which computation of unauthenticated session keys requires one round and proceeds with minimal computation and communication overhead. The first one-round authenticated 3KE version that preserved the unique efficiency properties of the original protocol and strengthened its security towards resilience against leakage of ephemeral (session-dependent) secrets was proposed recently by Manulis, Suzuki, and Ustaoglu (ICISC 2009). In this work we explore sufficient conditions for building such protocols. We define a set of admissible polynomials and show how their construction generically implies 3KE protocols with the desired security and efficiency properties. Our result generalizes the previous 3KE protocol and gives rise to many new authenticated constructions, all of which enjoy forward secrecy and resilience to ephemeral key-leakage under the gap Bilinear Diffie-Hellman assumption in the random oracle model. © 2012 Springer-Verlag.